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The vertex of a parabola is a crucial concept in understanding quadratic functions. It represents the highest or lowest point on the graph. For a quadratic function in standard form, \(f(x) = ax^2 + bx + c\), the vertex can be found using a simple formula:

The x-coordinate of the vertex is given by \(x = -\frac{b}{2a}\). By substituting the values of \(a\) and \(b\), you can find this coordinate. For the function \(g(x) = x^2 + 7x - 8\), we identified that \(a = 1\) and \(b = 7\).

Plugging these into the formula, we get:
\(x = -\frac{7}{2(1)} = -\frac{7}{2}\).

Next, substitute \(x = -\frac{7}{2}\) back into the function to find the y-coordinate:
\( g\left(-\frac{7}{2}\right) = \left(-\frac{7}{2}\right)^2 + 7\left(-\frac{7}{2}\right) - 8 = -\frac{81}{4}\).

So, the vertex of the parabola \(g(x)\) is at \(\left(-\frac{7}{2}, -\frac{81}{4}\right)\).

The axis of symmetry of a parabola is a vertical line that divides the graph into two mirror images. It passes through the vertex of the parabola. For a quadratic function in the standard form \(f(x) = ax^2 + bx + c\), you can find the axis of symmetry using the same formula as the x-coordinate of the vertex: \(x = -\frac{b}{2a}\).

Given our function \(g(x) = x^2 + 7x - 8\), the axis of symmetry is:
\(x = -\frac{7}{2(1)} = -\frac{7}{2}\).

Therefore, the axis of symmetry is \(x = -\frac{7}{2}\). This line helps us understand how the parabola is oriented on the graph and is useful for plotting additional points symmetrically around the vertex.

In a quadratic function, the minimum or maximum value depends on the direction in which the parabola opens. If the parabola opens upwards (\(a > 0\)), the vertex represents the minimum value. If it opens downwards (\(a < 0\)), the vertex is the maximum value.

Since the coefficient \(a = 1\) in our function \(g(x) = x^2 + 7x - 8\) is positive, the parabola opens upwards. This means there is a minimum value at the vertex.

The minimum value is the y-coordinate of the vertex. From our calculation, the vertex is \(\left(-\frac{7}{2}, -\frac{81}{4}\right)\), and thus the minimum value is \(y = -\frac{81}{4}\).

Understanding this helps us know the lowest point the function can reach, which is crucial for graphing and solving real-life problems.

Graphing a quadratic function involves plotting its key features: vertex, axis of symmetry, and additional points. For the function \(g(x) = x^2 + 7x - 8\), we start by noting the main characteristics:

• Vertex: \(\left(-\frac{7}{2}, -\frac{81}{4}\right)\)
• Axis of symmetry: \(x = -\frac{7}{2}\)
• Direction: Parabola opens upwards since \(a = 1\) is positive

Steps to graphing:

• First, plot the vertex \(\left(-\frac{7}{2}, -\frac{81}{4}\right)\) on the coordinate plane.
• Draw the axis of symmetry as a vertical line through the vertex at \(x = -\frac{7}{2}\).
• Choose additional x-values around the vertex and calculate their corresponding y-values to get more points.
• Plot these points and use the symmetry of the parabola to reflect them across the axis of symmetry.
• Connect the points with a smooth U-shaped curve.

This visual representation provides a deeper understanding of how the function behaves and aids in interpreting solutions.